Simple Polar Inequalities Exercises

Example 1

Write inequalities for r and θ describing the given region.

PICTURE polar ineqs 6 without pink

Answer

We have $\frac{π}{4}≤ θ ≤ \frac{3π}{4}$:PICTURE polar ineqs 6 with pink
Within those values of θ the value of r can be anything nonnegative, the appropriate bound is
0≤ r.
It would be possible to describe this region in a way that allowed r to be negative, but there's no need to
go to the extra trouble.

Example 2

Write inequalities for r and θ describing the given region.

PICTURE polar ineqs 7

Answer

The value of r can be anything greater than 2, therefore
2≤ r.
Since the region wraps all the way around, bounds for θ are optional. We could say
0≤ θ≤ 2π
or we could decline to give any bounds for θ.

Example 3

Write inequalities for r and θ describing the given region.

PICTURE polar ineqs 8

Answer

Counting the rays and the rings, we see that\frac{7π}{6}≤θ≤\frac{4π}{3}
and
2≤ r≤ 3.
PICTURE polar ineqs 9
{enumerate}

Example 4

Sketch the region described by the given inequalities.

$π ≤ θ ≤ \frac{3π}{2}$

Answer

Since no bounds are given for r, this region will extend outward infinitely. The conditions on θ include all of the third quadrant.
PICTURE: graph region

Example 5

Sketch the region described by the given inequalities.

r < 4

Answer

The condition r < 4 describes the inside of a circle of radius 4, not including the boundary of the circle. Since no bounds are given for θ
the region wraps all the way around the origin.
PICTURE: graph region r < 4.

Example 6

Sketch the region described by the given inequalities.

\frac{11&pi;}{6}&le; &theta; &le; 2&pi; and 2&le; r &le; 5

Answer

The conditions on θ describe an infinite pizza-slice-like wedge lying below the positive x-axis:
PICTURE: graph infinite pizza wedge described by $\frac{11π}{6}≤ θ ≤ 2π$
The conditions on r say we want to take a pizza-crust-shaped portion of that wedge:
PICTURE: graph region